A Symmetric Fractional-order Reduction Method for Direct Nonuniform Approximations of Semilinear Diffusion-wave Equations

Abstract

We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations. By using the novel order reduction method, the governing problem is transformed to an equivalent coupled system, where the explicit orders of time-fractional derivatives involved are all $\alpha /2$ $(1<\alpha <2)$. The linearized L1 scheme and Alikhanov scheme are then proposed on general time meshes. Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for the two kinds of difference schemes by $H^2$ energy method. An adaptive time stepping strategy which based on the (fast linearized) L1 and Alikhanov algorithms is designed for the semilinear diffusion-wave equations. Numerical examples are provided to confirm the accuracy and efficiency of proposed algorithms.

6. Appendix: Truncation Error Analysis

According to [17, Lemma 3.8 and Theorem 3.9], we have the follow lemma on estimating the time weighted approximation.

Lemma 6.1

Assume that $g\in C^2((0,T])$ and there exists a constant $C_g>0$ such that

$$\begin{aligned} |g''(t)|\le C_g(1+t^{\sigma -2}), \quad 0<t\le T, \end{aligned}$$

where $\sigma \in (0,1)\cup (1,2)$ is a regularity parameter. Denote the local truncation error of $g^{n-\vartheta }$ (here $\vartheta =\beta /2$) by

$$\begin{aligned} {\tilde{{\mathcal {R}}}}^{n-\vartheta }=g(t_{n-\vartheta })-g^{n-\vartheta }, \quad 1\le n\le N. \end{aligned}$$

If the mesh assumption MA holds, then

$$\begin{aligned} \sum _{j=1}^n P_{n-j}^{(n)}|{\tilde{{\mathcal {R}}}}^{j-\vartheta }|\le C_g\left( \tau _1^{\sigma +\beta }/\sigma +t_n^\beta \max _{2\le k\le n} t_{k-1}^{\sigma -2}\tau _k^2 \right) \le C\tau ^{\min \{\gamma \sigma ,2\}}. \end{aligned}$$

The following lemma is provided to analyze $(\mathcal{T}_f)_h^{n-\theta }$, which is analogous to Lemma 3.4 in [20].

Lemma 6.2

Assume that $\eta \in C([0,T])\cap C^2((0,T])$ and there exists a constant $C_u>0$ such that

$$\begin{aligned} |\eta ^{(k)}(t)|\le C_u(1+t^{\sigma -k}), \quad 0<t\le T,\quad k=1,2, \end{aligned}$$

where $\sigma \in (0,1)\cup (1,2)$ is a regularity parameter. Assume further that the nonlinear function $f(u,x,t)\in C^4({{\mathbb {R}}})$ with respect to u. Denote $\eta ^n=\eta (t_n)$ and the local truncation error

$$\begin{aligned} \mathcal{R}_f^{n-\theta }:=f(\eta (t_{n-\theta }),x,t)-\left[ f(\eta ^{n-1},x,t)+(1-\theta )f'_u(\eta ^{n-1},x,t)\nabla _\tau \eta ^n\right] . \end{aligned}$$

If the assumption MA holds, then

$$\begin{aligned} \sum _{j=1}^n P_{n-j}^{(n)}|{{\mathcal {R}}}_f^{j-\theta }|\le \left\{ \begin{array}{ll} C\tau ^{\min \{2\gamma \sigma ,2\}},\quad \theta =0;\\ C\tau ^{\min \{\gamma \sigma ,2\}},\quad \theta =\frac{\beta }{2}; \end{array} \quad 1\le n\le N. \right. \end{aligned}$$

Proof

Denote $\mathcal{R}_\eta ^{n-\theta }:=\eta (t_{n-\theta })-\eta ^{n-\theta }$. We have ${{\mathcal {R}}}_\eta ^{n-\theta }=0$ while $\theta =0$. By the Taylor expansion, we have

$$\begin{aligned} {{\mathcal {R}}}_f^{n-\theta } =&f_u'(\eta ^{n-1}){{\mathcal {R}}}_\eta ^{n-\theta }\\&+\left( (1-\theta )\nabla _\tau \eta ^n+\mathcal{R}_\eta ^{n-\theta }\right) ^2\int _0^1f''_u\left( \eta ^{n-1}+s (\eta (t_{n-\theta })-\eta ^{n-1}),x,t\right) (1-s)\,\mathrm {d}s. \end{aligned}$$

Following the proof of [20, Lemma 3.4] and using Lemma 6.1, the desired result holds immediately. $\square$

For $\mathbf{x}\in \Omega$, let $\xi ^n(\mathbf{x})$ be a spatially continues function and denote $\xi ^n_h:=\xi ^n(\mathbf{x}_h)$. One may apply the Taylor expansion to get

$$\begin{aligned} \Delta _h \xi ^n_h=&\int _0^1\left[ \partial _{xx}\xi ^n(x_i-sh_x,y_j)+\partial _{xx}\xi ^n(x_i+sh_x,y_j) \right] (1-s)\,\mathrm {d}s\\&+\int _0^1\left[ \partial _{yy}\xi ^n(x_i,y_j-sh_y)+\partial _{yy}\xi ^n(x_i,y_j+sh_y) \right] (1-s)\,\mathrm {d}s, \quad 1\le n\le N. \end{aligned}$$

Then we define a function ${{\mathcal {T}}}_f^{n}(\mathbf{x})$ by $(\mathcal{T}_f)_h^{n}={{\mathcal {T}}}_f^{n}(\mathbf{x}_h)$. If the assumptions in (1.5) and MA are satisfied, by Lemma 6.2 and the differential formula of composite function, we can obtain

$$\begin{aligned} \sum _{j=1}^n P_{n-j}^{(n)}\Vert \Delta _h({{\mathcal {T}}}_f)^{n-\theta }\Vert \le \left\{ \begin{array}{ll} C\tau ^{\min \{2\gamma \sigma _1,2\}},\quad \theta =0;\\ C\tau ^{\min \{\gamma \sigma _1,2\}},\quad \theta =\frac{\beta }{2}; \end{array} \quad 1\le n\le N. \right. \end{aligned}$$

(6.1)

For the spatial error, based on the regularity condition, it is easy to know that

$$\begin{aligned} \Vert {{\mathcal {S}}}^{n}\Vert \le C_uh^2, \quad 1\le n\le N. \end{aligned}$$

(6.2)

Then, by using (3.9),

$$\begin{aligned} \sum _{j=1}^n P_{n-j}^{(n)} \Vert ({{\mathcal {D}}}_\tau ^{\beta }\mathcal{S})^{j}\Vert \le \sum _{j=1}^n P_{n-j}^{(n)}\sum _{k=1}^jA_{j-k}^{(j)}\Vert \nabla _\tau \mathcal{S}^{k}\Vert&{=\sum _{k=1}^n\Vert \nabla _\tau \mathcal{S}^{k}\Vert \sum _{j=k}^nP_{n-j}^{(n)}A_{j-k}^{(j)}}\nonumber \\&=\sum _{k=1}^n\Vert \nabla _\tau {{\mathcal {S}}}^{k}\Vert \le C_u(1+t_{n}^{\sigma _1-1})h^2. \end{aligned}$$

(6.3)

We now consider the temporal truncation errors $(\mathcal{T}_{v1})_h^{n-\theta }$, $({{\mathcal {T}}}_{w})_h^{n-\theta }$, $(\mathcal{T}_{u})_h^{n-\theta }$ and $({{\mathcal {T}}}_{v2})_h^{n-\theta }$ in two situations: $\theta =0$ and $\theta =\beta /2$.

For a function g(t), define the global error

$$\begin{aligned} {{\mathcal {R}}}^{n-\theta }:=({{\mathcal {D}}}_t^\beta g)(t_{n-\theta })-({{\mathcal {D}}}_{\tau }^\beta g)^{n-\theta },\quad 1\le n\le N. \end{aligned}$$

For L1 approximation ($\theta =0$): We have $(\mathcal{T}_w)_h^{n}=({{\mathcal {T}}}_{v2})_h^{n}=0$ in this situation.

According to [15, Lemma 3.3] and [20, Lemma 3.3], the global consistency error of the L1 approximation can be presented in the following lemma.

Lemma 6.3

Assume that $g\in C^2((0,T])$ and there exists a constant $C_g>0$ such that

$$\begin{aligned} |g''(t)|\le C_g(1+t^{\sigma -2}), \quad 0<t\le T, \end{aligned}$$

where $\sigma \in (0,1)\cup (1,2)$ is a regularity parameter. If the assumption MA holds, it follows that

$$\begin{aligned} \sum _{j=1}^n P_{n-j}^{(n)}|{{\mathcal {R}}}^j|\le C_g\left( \tau _1^\sigma /\sigma +\frac{1}{1-\beta }\max _{2\le k\le n}(t_k-t_1)^\beta t_{k-1}^{\sigma -2}\tau _k^{2-\beta } \right) \le C\tau ^{\min \{2-\beta ,\gamma \sigma \}}. \end{aligned}$$

Define the functions ${{\mathcal {T}}}_{v1}^{n}(\mathbf{x})$ and $\mathcal{T}_u^{n}(\mathbf{x})$ by $({{\mathcal {T}}}_{v1})_h^n:={{\mathcal {T}}}_{v1}^{n}(\mathbf{x}_h)$ and $({{\mathcal {T}}}_{u})_h^n:={{\mathcal {T}}}_u^{n}(\mathbf{x}_h)$ respectively. Using similar techniques for (6.1), and Lemma 6.3 with the in assumptions (1.5) and MA, we have

$$\begin{aligned} \sum _{j=1}^n P_{n-j}^{(n)}\Vert \Delta _h{{\mathcal {T}}}_{v1}^{n}\Vert \le C\tau ^{\min \{2-\beta ,\gamma \sigma _2\}} \quad \text{ and } \quad \sum _{j=1}^n P_{n-j}^{(n)}\Vert \Delta _h{{\mathcal {T}}}_{u}^{n}\Vert \le C\tau ^{\min \{2-\beta ,\gamma \sigma _1\}}. \end{aligned}$$

(6.4)

For Alikhanov approximation ($\theta =\beta /2$):

The global consistency error estimate of the Alikhanov approximation is estimated in the next lemma.

Lemma 6.4

([17, Lemma 3.6]) Assume that $g\in C^3((0,T])$ and there exists a constant $C_g>0$ such that

$$\begin{aligned} |g'''(t)|\le C_g(1+t^{\sigma -3}), \quad 0<t\le T, \end{aligned}$$

where $\sigma \in (0,1)\cup (1,2)$ is a regularity parameter. Then

$$\begin{aligned} \sum _{j=1}^n P_{n-j}^{(n)}|{{\mathcal {R}}}^{j-\theta }|\le C_g\left( \tau _1^\sigma /\sigma +t_1^{\sigma -3}\tau _2^3+\frac{1}{1-\beta }\max _{2\le k\le n}t_k^\beta t_{k-1}^{\sigma -3}\tau _k^3/\tau _{k-1}^\beta \right) . \end{aligned}$$

By Lemmas 6.4, 6.1, the assumptions in (1.5) and MA, it is easy to get that

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)} \Vert \Delta _h({{\mathcal {T}}}_{v1})_h^{j-\theta }\Vert \le C\left( \tau _1^{\sigma _2}+\tau _2^3\tau _1^{\sigma _2-3}+\max _{2\le k\le n} (t_k-t_1)^{\beta }t_{k-1}^{\sigma _2-3}\tau _k^{3-\beta } \right) \le C\tau ^{\min \{3-\beta ,\gamma \sigma _2\}}, \end{aligned}$$

(6.5)

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)} \Vert \Delta _h({{\mathcal {T}}}_u)_h^{j-\theta }\Vert \le C\left( \tau _1^{\sigma _1}+\tau _2^3\tau _1^{\sigma _1-3}+\max _{2\le k\le n} (t_k-t_1)^{\beta }t_{k-1}^{\sigma _1-3}\tau _k^{3-\beta } \right) \le C\tau ^{\min \{3-\beta ,\gamma \sigma _1\}}, \end{aligned}$$

(6.6)

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)} \Vert \Delta _h({{\mathcal {T}}}_w)_h^{j-\theta }\Vert \le C\left( \tau _1^{\sigma _1+\beta }+\max _{2\le k\le n} t_{k-1}^{\sigma _1-2}\tau _k^2 \right) \le C\tau ^{\min \{2,\gamma \sigma _1\}}, \end{aligned}$$

(6.7)

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)} \Vert \Delta _h({{\mathcal {T}}}_{v2})_h^{j-\theta }\Vert \le C\left( \tau _1^{\sigma _2+\beta }+\max _{2\le k\le n} t_{k-1}^{\sigma _2-2}\tau _k^2 \right) \le C\tau ^{\min \{2,\gamma \sigma _2\}}. \end{aligned}$$

(6.8)